Answer:
Bob has owed this land for 21.68 years.
Step-by-step explanation:
The price of the land can be modeled by the following function:
[tex]P(t) = P(0)(1+r)^{t}[/tex]
In which P(t) is the price after t years, P(0) is the initial price, and r is the growth rate, as a decimal.
In this problem, we have that:
We want to find t, when [tex]P = 46017, P(0) = 15990, r = 0.05[/tex]. So
[tex]P(t) = P(0)(1+r)^{t}[/tex]
[tex]46017 = 15990(1+0.05)^{t}[/tex]
[tex](1.05)^{t} = \frac{46017}{15990}[/tex]
[tex](1.05)^{t} = 2.88[/tex]
We have that:
[tex]\log{a^{t}} = t\log{a}[/tex]
So to find t, we apply log to both sides
[tex]\log{(1.05)^{t}} = \log{2.88}[/tex]
[tex]t\log{1.05} = \log{2.88}[/tex]
[tex]t = \frac{\log{2.88}}{\log{1.05}}[/tex]
[tex]t = 21.68[/tex]
Bob has owed this land for 21.68 years.
